Spitzer's formula
In probability theory, Spitzer's formula or Spitzer's identity gives the joint distribution of partial sums and maximal partial sums of a collection of random variables. The result was first published by Frank Spitzer in 1956.[1] The formula is regarded as "a stepping stone in the theory of sums of independent random variables".[2]
Statement of theorem
Let X1, X2, ... be independent and identically distributed random variables and define the partial sums Sn = X1 + X2 + ... + Xn. Define Rn = max(0,S1,S2,...,Sn). Then[3]
![{\displaystyle \sum _{n=0}^{\infty }\phi _{n}(\alpha ,\beta )t^{n}=\exp \left[\sum _{n=1}^{\infty }{\frac {t^{n}}{n}}\left(u_{n}(\alpha )+v_{n}(\beta )-1\right)\right]}](../I/m/88baff5fa1bb3e1c9e79d316a0c813bcf961a565.svg)
where
![{\displaystyle {\begin{aligned}\phi _{n}(\alpha ,\beta )&=\operatorname {E} (\exp \left[i(\alpha R_{n}+\beta (R_{n}-S_{n})\right])\\u_{n}(\alpha )&=\operatorname {E} (\exp \left[i\alpha S_{n}^{+}\right])\\v_{n}(\beta )&=\operatorname {E} (\exp \left[i\beta S_{n}^{-}\right])\end{aligned}}}](../I/m/e9e2d1183da987cee0d0c564d5c1e4674045bb13.svg)
and S± denotes (|S| ± S)/2.
Proof
Two proofs are known, due to Spitzer[1] and Wendel.[3]
References
- 1 2 Spitzer, F. (1956). "A combinatorial lemma and its application to probability theory". Transactions of the American Mathematical Society. 82 (2): 323–339. doi:10.1090/S0002-9947-1956-0079851-X.
- ↑ Ebrahimi-Fard, K.; Guo, L.; Kreimer, D. (2004). "Spitzer's identity and the algebraic Birkhoff decomposition in pQFT". Journal of Physics A: Mathematical and General. 37 (45): 11037. doi:10.1088/0305-4470/37/45/020.
- 1 2 Wendel, James G. (1958). "Spitzer's formula: A short proof". Proceedings of the American Mathematical Society. 9 (6): 905–908. doi:10.1090/S0002-9939-1958-0103531-2. MR 0103531.
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